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a458723 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 | import Mathlib
import OAI.Analysis.Conductivity.Variational.CompletedPotential
import OAI.Analysis.Conductivity.Variational.SynchronizedChartWaves
import OAI.Analysis.Conductivity.Variational.RepairPushedGraphStability
namespace OAI
noncomputable section
open MeasureTheory
open scoped ENNReal
open Matrix Filter Topology
open Set MeasureTheory Filter Topology
open scoped BigOperators
open Set MeasureTheory Filter Topology
open scoped Manifold
open Set Filter
open scoped Topology
open Set Filter MeasureTheory
open scoped Topology Manifold ENNReal
open Set
namespace ScalarConductivity
open Matrix Set Filter Topology MeasureTheory
open scoped Matrix.Norms.Elementwise
def splitJacobian (d : Coord3) (L : Coord3 →L[ℝ] ℝ) (c z : ℝ) : Mat3 :=
operatorMatrix (ContinuousLinearMap.id ℝ Coord3 + lineShiftGradientOperator d c (z • L))
lemma splitJacobian_continuous (d : Coord3) (L : Coord3 →L[ℝ] ℝ) (c : ℝ) :
Continuous (splitJacobian d L c) :=
operatorMatrix_continuous.comp (continuous_const.add
((lineShiftGradientOperator d c).continuous.comp (continuous_id.smul continuous_const)))
lemma matrix_two_pairing_symm (E F : Matrix (Fin 3) (Fin 2) ℝ)
(h : (Eᵀ * F) 1 0 = (Eᵀ * F) 0 1) : (Eᵀ * F).IsSymm := by
ext i j
fin_cases i <;> fin_cases j <;> simp_all [Matrix.transpose_apply]
theorem exists_local_two_field_split
(μ : Measure Coord3) [μ.IsAddHaarMeasure]
(u : Coord3 → Fin 2 → ℝ) (hu : ContDiff ℝ (↑(⊤ : ℕ∞)) u)
(A : Coord3 → Symmetric3) {p : Coord3} (hA : ContinuousAt A p)
(hD : Function.Surjective (fderiv ℝ u p))
(d : Coord3) (L : Coord3 →L[ℝ] ℝ) (hLd : L d = 1)
(B : Mat3) (hB : B.IsSymm) (hBn : ∀ v, L (B *ᵥ v) = 0)
{a b c m : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (ha1 : a ≤ 1) (hb1 : b ≤ 1)
(hm : 0 < m) (hpath : ∀ z ∈ Icc (-a) b, m ≤ 1 + c * z)
(C : ℝ → Symmetric3) (hC : ∀ z ∈ Icc (-a) b, ContinuousAt C z)
(hdet : ∀ z ∈ Icc (-a) b, 0 < (splitJacobian d L c z).det)
(hconstit : ∀ z ∈ Icc (-a) b,
(C z).val = (splitJacobian d L c z).det⁻¹ •
(splitJacobian d L c z * ((A p).val + z • B) * (splitJacobian d L c z)ᵀ))
{G : Set (ℝ × Symmetric3)} (hG : IsOpen G)
(hCG : ∀ z ∈ Icc (-a) b, (z, C z) ∈ G)
{U : Set Coord3} (hU : IsOpen U) (hpU : p ∈ U) :
∃ (X : OpenPartialHomeomorph Coord3 Coord3) (V : Set Coord3),
IsOpen V ∧ p ∈ V ∧ V ⊆ U ∧ V ⊆ X.source ∧
ContDiff ℝ (↑(⊤ : ℕ∞)) X ∧
ContDiffOn ℝ (↑(⊤ : ℕ∞)) X.symm X.target ∧
∃ M : ℝ, 0 < M ∧
∀ H : SmoothScalar ℝ, Function.Periodic (smoothDirection 1 H).val 1 →
(∫ t in (0 : ℝ)..1, (smoothDirection 1 H).val t = 0) →
(∃ B : ℝ, ∀ t, |H.val t| ≤ B) →
(∀ t, -a ≤ (smoothDirection 1 H).val t ∧ (smoothDirection 1 H).val t ≤ b) →
∀ χ : SmoothScalar Coord3, HasCompactSupport χ.val → tsupport χ.val ⊆ X '' V →
(∀ x, 0 ≤ χ.val x ∧ χ.val x ≤ 1) →
∃ F : ℝ → Fin 2 → Coord3 → Coord3,
(∀ k j, ContDiff ℝ (↑(⊤ : ℕ∞)) (F k j)) ∧
(∀ k j, HasCompactSupport (F k j) ∧ tsupport (F k j) ⊆ V ∧
tsupport (F k j) ⊆ X.symm '' tsupport χ.val) ∧
(∀ k j (ψ : Coord3 → ℝ), ContDiff ℝ (↑(⊤ : ℕ∞)) ψ →
(∫ y, fderiv ℝ ψ y (F k j y) ∂μ) = 0) ∧
∀ᶠ k : ℝ in atTop, ∃ Y : Coord3 ≃ Coord3,
ContDiff ℝ (↑(⊤ : ℕ∞)) Y ∧ ContDiff ℝ (↑(⊤ : ℕ∞)) Y.symm ∧
(∀ x ∉ V, Y x = x) ∧ Y '' V = V ∧
(∀ x, Y x = x + (c * k⁻¹ * localPullback X χ.val x *
H.val (k * L (X x))) • d) ∧
(∀ x, |(fderiv ℝ Y x).det| ≤ M) ∧
∀ x ∈ V,
let z := localPullback X χ.val x * (smoothDirection 1 H).val (k * L (X x))
let E := gradientColumns (fderiv ℝ u x)
let Ft : Matrix (Fin 3) (Fin 2) ℝ := (A x).val * E + Matrix.of (fun i j => F k j x i)
let J := operatorMatrix (fderiv ℝ Y x)
let At := repairPushed (C z) J E Ft
0 < J.det ∧ LinearIndependent ℝ E.col ∧ (z, At) ∈ G ∧
At.val * gradientColumns (fderiv ℝ (u ∘ Y.symm) (Y x)) = pushFlux J Ft := by
classical
let D := fderiv ℝ u p
let E₀ := gradientColumns D
let E : Coord3 → Matrix (Fin 3) (Fin 2) ℝ := fun y => gradientColumns (fderiv ℝ u y)
let F₀ := (A p).val * E₀
let Fe : Coord3 → Matrix (Fin 3) (Fin 2) ℝ := fun y => (A y).val * E y
let J := splitJacobian d L c
let Fb : ℝ → Matrix (Fin 3) (Fin 2) ℝ := fun z => ((A p).val + z • B) * E₀
have hEc : Continuous E := by
apply continuous_pi; intro i
apply continuous_pi; intro j
exact (continuous_apply j).comp ((hu.continuous_fderiv (by simp)).clm_apply continuous_const)
have hFc : ContinuousAt Fe p := matrix_mul_contAt (continuousAt_subtype_val.comp hA) hEc.continuousAt
have hFbc : Continuous Fb := by
apply continuous_iff_continuousAt.mpr
intro z
exact matrix_mul_contAt (continuousAt_const.add (continuousAt_id.smul continuousAt_const)) continuousAt_const
have hrank : LinearIndependent ℝ E₀.col := gradientColumns_rank D hD
have hexact : ∀ z ∈ Icc (-a) b, repairPushed (C z) (J z) E₀ (Fb z) = C z := by
intro z hz
exact repairPushed_exact (C z) (J z) ((A p).val + z • B) E₀ (hdet z hz).ne' (hconstit z hz)
obtain ⟨O, _, hKO, ε, hε, heps⟩ := repairPushed_graph_stability isCompact_Icc
C J (fun _ => E₀) Fb hC (fun _ _ => (splitJacobian_continuous d L c).continuousAt)
(fun _ _ => continuousAt_const) (fun _ _ => hFbc.continuousAt) hdet (fun _ _ => hrank) hG
(fun z hz => by rw [hexact z hz]; exact hCG z hz)
have hnb : ∀ᶠ x in 𝓝 p, ‖E x - E₀‖ < ε/2 ∧ ‖Fe x - F₀‖ < ε/2 := by
have he := hEc.continuousAt.eventually (Metric.ball_mem_nhds (E p) (half_pos hε))
have hf := hFc.eventually (Metric.ball_mem_nhds (Fe p) (half_pos hε))
simpa only [Metric.mem_ball, dist_eq_norm] using he.and hf
obtain ⟨W, hWb, hWo, hpW⟩ := mem_nhds_iff.mp hnb
obtain ⟨X, hpX, hXs, hX, hXi, hXu, hDX⟩ := exists_potential_chart hu p hD
(hU.inter hWo) ⟨hpU, hpW⟩
have hpol := symmetric_wave_polarization D B hB L hBn
obtain ⟨V, hV, hpV, hVs, hpert⟩ := exists_synchronized_chart_waves
(by simp [Coord3]) μ X hX hXi D hD u (fun y _ => hXu y)
hpX hDX d L hLd ha hb ha1 hb1 hm (by positivity : 0 < ε/2)
hpath (B * E₀).col hpol.1 hpol.2
obtain ⟨M, hM, hMbd⟩ := synchronized_det_bound d c L (ε/2)
refine ⟨X, V, hV, hpV, hVs.trans (fun _ hx => (hXs hx).1), hVs, hX, hXi, M, hM, ?_⟩
intro H hHp hHm hH hHr χ hχ hχV hχb
obtain ⟨F, hF, hFs, hsym, hdiv, hlim⟩ := hpert H hHp hHm hH hHr χ hχ hχV hχb
refine ⟨F, hF, hFs, hdiv, ?_⟩
filter_upwards [hlim] with k hk
obtain ⟨Y, hY, hYi, hoff, himage, hform, hFerr, hJerr⟩ := hk
refine ⟨Y, hY, hYi, hoff, himage, hform, ?_, ?_⟩
· intro x
apply hMbd _ _ _ (hJerr x).le
have hz := scalar_cutoff_segment ha hb (localPullback_bounds X χ.val hχb x) (hHr (k * L (X x)))
exact abs_le.mpr ⟨by linarith only [hz.1, ha1], by linarith only [hz.2, hb1]⟩
intro x hx
let z := localPullback X χ.val x * (smoothDirection 1 H).val (k * L (X x))
let Q : Matrix (Fin 3) (Fin 2) ℝ := fun i j => F k j x i
let Jy := operatorMatrix (fderiv ℝ Y x)
have hz : z ∈ Icc (-a) b := scalar_cutoff_segment ha hb
(localPullback_bounds X χ.val hχb x) (hHr _)
have hsmall := hWb (hXs (hVs hx)).2
have hdJ : ‖Jy - J z‖ < ε/2 := by
change ‖operatorMatrix (fderiv ℝ Y x) - operatorMatrix _‖ < ε/2
rw [← operatorMatrix_sub]
exact (operatorMatrix_norm_le _).trans_lt (hJerr x)
have hdF : ‖Fe x + Q - Fb z‖ < ε := by
have hq : ‖Q - z • (B * E₀)‖ < ε/2 := by
apply columns_norm_lt _ (half_pos hε)
intro j
exact hFerr j x
have heq : Fe x + Q - Fb z = (Fe x - F₀) + (Q - z • (B * E₀)) := by
dsimp only [Fb, F₀]
rw [Matrix.add_mul, Matrix.smul_mul]
abel
rw [heq]
exact (norm_add_le _ _).trans_lt (by linarith only [hq, hsmall.2])
have hbound : ‖(Jy - J z, E x - E₀, Fe x + Q - Fb z)‖ < ε := by
rw [Prod.norm_def, Prod.norm_def, max_lt_iff, max_lt_iff]
exact ⟨hdJ.trans (by linarith), hsmall.1.trans (by linarith), hdF⟩
have hs := heps z (hKO hz) (Jy - J z) (E x - E₀) (Fe x + Q - Fb z) hbound
simp only [add_sub_cancel] at hs
refine ⟨hs.1, hs.2.1, hs.2.2, ?_⟩
rw [gradientColumns_recomposition Y (hY.differentiable (by simp))
(hYi.differentiable (by simp)) u (hu.differentiable (by simp))]
apply repairPushed_mul _ _ _ _ hs.1.ne' hs.2.1
rw [Matrix.mul_add]
apply (symmetric_gram_pairing (A x) (E x)).add
apply matrix_two_pairing_symm
rw [← gradientColumns_pairing, ← gradientColumns_pairing]
exact hsym k x
end ScalarConductivity
end
end OAI
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